Figure REF (c1) represents type C where only L depends on its present.
In this case, a significant amount of \(\sigma _{\rm L\rightarrow F}\)  appears
due to the dependence of the present state of both L and F on the past state
of L, as shown in Fig. REF (c2). L imparts information from its history
on to the future dynamics of F, and meanwhile, this information is already contained
in the present state of F due to the present state of F's
dependence on that same history. In contrast to types A and B, there is a significant amount of
shared history between F and L, as shown in Fig. REF (c2). As in the binary
system proposed by Schreiber [1]} having
the same graph representation as Fig.REF (c1), there
exists a significant amount of \(\mathcal {M}_{\rm F \rightarrow L}\)  even though there is no direct interaction
in that direction. By decomposing \(\mathcal {M}_{\rm F \rightarrow L}\)  into \(\mathcal {I}_{\rm F \rightarrow L}\) 
and \(\sigma _{\rm L\rightarrow F}\) , we quantitatively show that the spurious amount of \(\mathcal {M}_{\rm F \rightarrow L}\)  is coming solely from
\(\sigma _{\rm L\rightarrow F}\)  and is thus not intrinsically coming from F. Transfer entropy
\(\mathcal {T}\)  was introduced to reconcile this issue [1]}. \(\mathcal {T}_{\rm F \rightarrow L}\) 
does in fact reduce the amount information flow in that direction in our model, given that
\(\mathcal {T}_{\rm F \rightarrow L}\)  is significantly less than \(\mathcal {M}_{\rm F \rightarrow L}\) 
in Fig. REF (c3).
